Experiment about electron speed distribution

The Maxwell-Boltzmann distribution of particle speeds in a gas has been verified with an experiment, where particles are allowed to exit from a container through a small hole, and the particle beam then hits a rotating cylinder with a slit. The speed distribution can be solved from the hit signs in the interior of the cylinder.

What about a similar experiment with electrons in a metal? If a metal is charged negatively, does it cause a small steady current of electrons flying out? Could the speed distribution of electrons in a metal be measured with a similar rotating cylinder trick? Has this experiment been carried out, or is there some difficulties that make the experiment impossible?

You can certainly measure energy distribution functions for electrons, for example those exiting a metal that has just been irradiated by a high-power laser. There are various ways to do this, including using a magnetic spectrometer. Trouble is, you aren't sampling the whole distribution, you're sampling those that managed to escape - and, those electrons will interact with each other directly and indirectly through their self-generated magnetic field, as well as with the surface they just left, so by the time you catch them their energy distribution has probably changed. Atoms in a gas are much easier to deal with because they aren't charged.

I could guess that one of the problems is that the more violently you get the electrons out, the more you disturb their original distribution. Anything that looks like a spark doesn't look very convincing bunch of electrons to observe.

If a piece of metal is charged negatively, and placed in vacuum, does the excess charge leak out slowly? To me that would look like something where the original distribution could be still solved from observations.